<p>If the tangent at \((1, 7)\) to the curve \(x^2 = y - 6\) touches the circle \(x^2 + y^2 + 16x + 12y + c = 0\), then the value of \(c\) is</p>
Step-by-Step Solution
Key Concept: Find the tangent line equation at the given point on the parabola, then use the condition that this line is tangent to the circle (distance from center equals radius) to determine c.
<p><strong>Step 1:</strong> Verify point lies on parabola: x² = y - 6 gives 1 = 7 - 6 = 1 ✓</p><p><strong>Step 2:</strong> Find tangent to parabola y = x² + 6 at (1,7). Using dy/dx = 2x, at x = 1: slope = 2. Tangent line: y - 7 = 2(x - 1), giving <strong>2x - y + 5 = 0</strong></p><p><strong>Step 3:</strong> Rewrite circle: x² + y² + 16x + 12y + c = 0 can be written as (x + 8)² + (y + 6)² = 100 - c. Center: (-8, -6), Radius: √(100 - c)</p><p><strong>Step 4:</strong> For tangency, distance from center to tangent line equals radius: |2(-8) - (-6) + 5|/√(4 + 1) = √(100 - c). This gives |-16 + 6 + 5|/√5 = √(100 - c), so |-5|/√5 = √(100 - c)</p><p><strong>Step 5:</strong> 5/√5 = √(100 - c) ⟹ √5 = √(100 - c) ⟹ 5 = 100 - c</p><p>∴ <strong>c = 95</strong></p>
Correct Answer: C